Samadhi's school help thread

[Lepton*]

I think that the best exams are overly long, homogeneous, comprehensive, and challenging. It forces the students to prove their knowledge. The GREs subject tests are written such that if you can correctly answer half of the questions, then you will have a very favourable score.

I wrote a stupid 20-minute exam (in a 1.5 hour period) today. I didn't feel like I was able to demonstrate my knowledge. I feel that I have demonstrated my knowledge after writing prize exams (Putnam, etc).

Education research, if I remember correctly, identifies a number of different levels of comprehension. Here is an overview. In essence, one should learn up to the ability to synthesize and critically evaluate similar arguments. In math, this could mean being able to construct a prove using previously-learned techniques (without having seen this particular proof before) and seeing where a faulty proof might break down. Almost nobody tests for these things because such questions are Hard and people complain. Thus we have a status quo where university examinations don't even test for what they should be testing for.

[CzarJ]

Ooh, the Putnam was fun.

I'm pretty sure I got a zero, but it was fun.

[casinopete]

I've experienced the Hard Test in a hardware class. I got a 36, and was absolutely mortified until it became clear that it scored a B+.

(This class was 15hrs+/week of work, despite being 4 credits)

[Samadhi]

Tests came back today. I tied for second highest. A 75 of 100. All but 12 lost points were from mistakes that I really shouldn't have made. 6 points were from not determining whether the function was differentiable, the other 6 points I'm pissed about.

The question was 'Let a = <2,1,3>, b = <1,3,5>, c = <1,0,6>; Are these vectors coplanar?'

I wrote:

[Samadhi]

I also took my Linear Algebra test today. I think I did worse than my calc test. Proofs! Arrrrgggh!

[austinap*]

I hate it when tests are easy. Test should be hard, otherwise what is there to actually differentiate between A students and students that can just reproduce an answer? I actually seem to do considerably worse in easy classes because the tests don't seem to test actual comprehension of the course material but instead the ability to jump through hoops, and I absolutely hate that. If I'm ever a professor, my tests are going to be insanely hard, but the curve is going to be great. Thats how tests should be. I garantee that I'll know which students really understand the material.

[Samadhi]

Yeah, but you'd think that out of 15 people, someone should get an A.

[MatthewV]

I did :-)

I personally feel that tests should also be open-book, open-note, use-anything-you-want. Recognizing how to use the tools and method of thinking you have learned are what I personally find to be important.

[Samadhi]

I'd be happy to post the test for people to take and report the results (on your honor taking only 1:50).

[Samadhi]

That is if anyone is interested.

[MatthewV]

Ouch- almost two hours for tests. My class is an hour each time (4 days a week). He has to make tests that are reasonable to finish in that time frame. This means that I finish in 30 minutes and then can spend 25 minutes working on one problem that doesn't make sense :-)

[Samadhi]

Let me PM you the test and see how you do. Shouldn't be hard. It's open internet, which is better than open book.

[The Ragin' South Asian]

take a math test in my free time? where do i sign up?!

[Samadhi]

I think it's key that time limits are imposed.....so if you are actually interested, PM me and I'll PM you the test.

[CzarJ]

I'd love to (as worthless as a human being as it makes me feel to admit it), but let me wait until after my calc test on Tuesday. I need all the study time I can get.

[Samadhi]

Ok, I'll send you a pm, you respond within 5 minutes and I'll send you the test. Time starts 5 minutes after I send.

[CzarJ]

Whaaat? I'm confused. Far too confused to take a test.

[Samadhi]

Or PM me with a confirmation time and we'll do the above.

[CzarJ]

How about Wednesday? =P

[Samadhi]

I'm just trying to set a time limit. Barring IM I don't know a better way to synchronize than through IM. (just trust me)

[Samadhi]

Whenever or never, your choice.

[CzarJ]

Alright, IM me on Wednesday. =P

[Samadhi]

You want the practice test?

[CzarJ]

(P.S., we could have done this all through PM, but as I'm sure you've noticed, threadjacking your thread is just about my favorite thing to do, so thanks for the diversion. =P)

Psh, I don't need no stinkin' practice test.

[Samadhi]

Yeah. That's what I thought.

[Samadhi]

Seriously though. Talking with everyone who has been taught by this guy (and there's a few, he's the only one at my 2 year that teaches Calc III or Linear Algebra) he's pretty unsympothetic. Not to say he isn't fair...everything on the tests were in the lectures.

But just imagine your worst homework questions..the ones that make you go ???....those are the questions on the test. Well, not so much calc III, but definitely Linear.

[CzarJ]

Hehe, I'm curious to see how I'll do without. My last Calc (IV, although the III ones weren't a whole lot better) test was a lot harder than the homework, which I definitely didn't expect.

[Samadhi]

I'm curious to see how you do too.

I'm about to go to bed though. Pm me tomorrow or whenever when you want to try out the test.

[CzarJ]

Yeah, I don't know what you were still doing up. I don't know what I was still doing up, either.

[Samadhi]

I'm getting a 'null pointer exception' on my most recent project. I'm not sure why. I'm going to do this without using an array, but if anyone can tell me why it's doing that, I'd be very happy.

The class used by my program
My program

[Samadhi]

Oh, I get it when I use the set method. It works fine when I don't use an array.

[CrystyB]

(disclaimer: i'm doing this in my head, i don't have access to Java right now) i think the reason is that

[Samadhi]

Perfect! Thanks

[MatthewV]

I wouldnt mind seeing the test but I doubt I would want to do it as a timed event.

(and I have my own test in a few hours...)

[Samadhi]

Here ya go.

1. If f(s,t,u) = s ² t ³ e ^{u^2st^3} then find all of its partial derivatives.
   
2. 1. Determine the distance from the plane 12x + 13y + 5z = 0 to the point (1, 1, -5)
      
   2. Let a = <2,1,3>, b = <1,3,5>, c = <1,0,6>; Are these vectors coplanar?
      
   3. Find parametric and vector equations of the line which passes through P ₁ (4,5,1) and P ₂ (2,1,3).
      
   4. Find the projection of vector a onto vector c where a and c are from part B above.
      
   5. Find the equation of the plane that contains the line r(t) = <-1,1,2> + t <3,2,4> and is perpendicular to the plane 2x + y - 3z + 4 = 0
   
   
3. 1. Show that f(x,y) =
      {x ² /((x ² + y ² )^ ^1/2 ) if (x,y) != (0,0)
      0 if (x,y) = (0,0)}
      is continuous at (0,0).
      
   2. Is f(x,y) differentiable at (0,0)?
   
   
4. Find two points on the surface z = 8 - x ² - 2y at which the tangent plane is parallel to the line x = 3 + 4t, y = 2t, z = 2 + t
   
5. Find the directional derivative of f(x,y) = 2xy - y/x at (1,2) in the direction of a vector v = <2,1>
   
6. Find D(f o g) if f(u, v) = (uv, u ² - v ³ , v - u) and g(x,y) =(u(x,y), v(x,y)) where u(x,y) = e ^x+y and v(x,y) = x+y at the point (x,y) = (2,1)
   
7. Find the equation of the tangent plane aqnd the normal vector to the following:
   1. 3xy ² + z ² = 10 at (2,1,-2)
      
   2. z = f(x,y) = ye ^{x^2 + y^2} at (3,2)
   
   
8. Find the equation for the tangent line and a normal vector to the curve defined by:
   
   1. x ² + y ² = 6 at (2,1)
      
   2. y = f(x) = 45/x ² at ((5^1/2)/9)

[Antrax]

1. It's not. I do nothing.
_________________
After years of disappointment with get rich quick schemes, I know I'm gonna get rich with this scheme. And quick!

[Samadhi]

Here's my linear algebra test.

1. Find the standard matrix representation A of the linear operator
   T: R ³ ----> R ³
   that rotates a vector 30 degrees counterclockwise about the z-axis and then dialates that vector by a factor of 2.
   
2. Show that 2 is an eigenvalue of the Linear operator T above and then find its corresponding eigenvector.


1. Let S = {v ₁ , v ₂ , v ₃ } where v ₁ = <2, 0, -1>, v ₂ = <1, 5, 0>, v ₃ = <0, 3, 1>, where B = {e ₁ , e ₂ , e ₃ } is the standard basis for R ³
   
   1. Show that S is a basis for R ³
      
   2. Find (u) _S where (u) _B = <7, 5, 2>
   
   
2. For the matrices shown, H is the reduced row echelon form that is row equivalent to A:
   
   
   1. express the solution of Ax = b as X = Xh + p with p satisfying Ap = b and Xh satisfies the corresponding homogeneous system.
      
   2. In the Ax = b system above, which variables are restricted and which are free?
      
   3. Prove that the set of all vectors x in R ⁴ satisfying the homogeneous system is a subspace of R ⁴ . What is a basis for this subspace? What is its dimension?
   
   
3. Determine a basis for the subspace of R ³ given by the plane 3x -2y + 5z = 0.
   
4. Let T: V ---> V' be a linear transformation. We know that the set of all v in V such that T(v) = 0, the zero vector in V, is a subspace of V. This subspace is called the kernel of the transformation T (or the vector space of solutions of transformations standard matrix homogeneous system).
   Show that if a linear transformation has only the zero subspace as its kernal then it is a one to one transformation.
   
5. Find the elementary matrices E _i so that the following matrix A can be written as R = E _n E _n-1 ....E ₁ A where R is the row reduced echelon form of A
   
   
6. Suppose that T: R ² ----> R ² given by
   
   Prove that T is one to one.
   
7. Evaluate det
   

   Code:

   1  3  5 4 14 12 9  2  0

   
   by reducing the matrix to row echelon fom.
   
8. Find the equation of the plane that contains the line L: <2 + 5t, 1-t, 2 +2t/3> and is perpendicular to the plane 2x + 11y + 3z = 5
   
9. Let S = {v ₁ , v ₂ , v ₃ } where v ₁ = <2, 0, 0>, v ₂ = <1, 2, -2>, v ₃ = <2, 1, 1>. Does S span R ³ ? Prove your answer.

[Samadhi]

[Antrax]

[Samadhi]

I feel *wooshed*